R. J. Warne has defined an wⁿ-right cancellative semigroup to be a right cancellative semigroup with identity whose ideal structure is order isomorphic to (I^o)ⁿ, where I^o is the set of non-negative integers and n is a natural number, under the reverse lexicographic order. Warne has described, modulo groups, the structure of such semigroups ["Bisimple Inverse Semigroups Mod Groups," Duke Math. J., Vol. 14 (1967), pp. 787-811]. He has used this structure and the theory of right cancellative semigroups having identity on which Green's relation J: is a congruence to describe the homomorphisms of an ωⁿ-right cancellative semigroup into an ωⁿ-right cancellative semigroup when 1 ≤ n ≤ 2 and m ≤ n ["Lectures in Semigroups," West Virginia Univ., unpublished]. We have described, modulo groups, the homomorphisms of an ωⁿ-right cancellative semigroup into an ω^m-right cancellative semigroup for arbitrary natural numbers n and m. One of the main results is the following: Theorem: Let P = (G ,(I^o)ⁿ , γ₁,...,γ_n, w₁,…,w_Ø(n)) and P^* = (G ,(I^o)ⁿ , α₁,...,α_n, t₁,…,t_Ø(n)) be ωⁿ-right cancellative semigroups where Ø(x) = ½x(x-1). Let z₁, ... ,z_n be elements of G^* and let f be a homomorphism of G into G^* such that (1) (Af)^(U_kg)C_{z_k} = (Aγ_kf) for A ∈ G where 1 ≤ k ≤ n and (2) ((z_k+s)^(U_kg)(U_kg)^(U_k+sg)C_{z_k} = w_Ø(n-k)+sf where 1 ≤ k ≤ n and 1 ≤ s ≤ n - k. The elements U_k (1 ≤ k ≤ n) are generators of (I^o)ⁿ, xC_{z_k} = z_kxz_k⁻¹ for x ∈ G^*, and x^a,a^b in G^* (x ∈ G^*; a,b ∈ (I^o)ⁿ are specified. Define, for (A,a₁,...,a_n) ∈ P, (A,a₁,...,a_n)M = [(Af)(a₁,...,a_n)h,(a₁,...,a_n)g] where h is a specified function from (I^o)ⁿ into G* and g is a determined endomorphism of (I^o)ⁿ. Then, M is a homomorphism of P into P* and every homomorphism of P into P* is obtained in this fashion. M is an isomorphism if and only if f and g are isomorphisms. M is onto when g is the identity and f is onto. Results similar to this theorem have been obtained when P* is an ω^m-right cancellative semigroup with m < n and m > n. Let I be the set of integers. Let S be a bisimple semigroup and let E_S denote the set of idempotents of S. S is called ωⁿ-bisimple if and only if E_S, under its natural order, is order isomorphic to I x (I^o)ⁿ under the reverse lexicographic order n ≥ 1. S is called I-bisimple if and only if E_S, under its natural order, is order isomorphic to I under the reverse usual order. Warne has described, modulo groups, the structure of ωⁿ-bisimple, ωⁿI-bisirnple and I-bisimple semigroups in ["Bisimple Inverse Semigroups Mod Groups," Duke Math. J., Vol. 14 (1967), pp. 787-811], ["ωⁿI-bisimple Semigroups," to appear], and ["I-bisimple Semigroups," Trans. Amer. Math. Soc., Vol. 130 (1968), pp. 367-386] respectively. We have described the homomorphisms of S into S* , by use of the homomorphism theory of ωⁿ-right cancellative semigroups, for the cases (i) S ωⁿ-bisimple and S* ω^m-bisimple and (ii) S I-bisimple or ωⁿI-bisimple and S* I-bisimple or ω^mI-bisimple where m and n are natural numbers. The homomorphisms of S onto S* are specified for cases (i) and (ii). Warne has determined the homomorphisms of S onto S* in certain of these cases as he studied the extensions and the congruences of ωⁿ-bisimple, ωⁿI-bisimple, and I-bisimple semigroups. Papers on these subjects are to appear at some later date. / Ph. D.

We give a method, by solving a nonlinear system of equations, for Gauss harmonic interpolation formulas which are useful for approximating, the solution of the Dirichlet problem. We also discuss approximations for integrals of the form I(f) = (1/2πi) ∫_L (f(z)/(z-α)) dz. Our approximations shall be of the form Q(f) = Σ_k=1ⁿ A_kf(τ_k). We characterize the nodes τ₁, τ₂, …, τ_n, to get the maximum precision for our formulas. Finally, we propose a general problem of approximating for linear functionals; our results need further development. / Ph. D.

The classical theorem of N. Wiener and P. Lévy states that if f(x) has an absolutely convergent Fourier series and W(z) is an analytic function whose domain contains the range of f(x), then W[f(x)] also has an absolutely convergent Fourier series. The main result of this paper is an analog of the Wiener-Lévy Theorem in which we consider analytic transformations acting upon kernels of integral operators of the form (Tφ)(x) = ∫_-π^π K(x,y)φ(y)dy and the so-called s-numbers of K and W[K] take the place of the classical Fourier coefficients of f and W[f]. Theorem: Let W(z) be analytic in a region R and let K(x,y) map the square [-π,π] x [-π,π] continuously into R. If {φ_n}_n=1^∞ and {ψ_n}_n=1^∞ are a full set of continuous singular functions for K(x,y) and ∑_n=1^∞ [s_n (K)]^P | | φ_n | |_∞^P| | ψ<sub>n</sub | |_∞^P < ∞ for some 0 < p ≤ 1, then ∑_n=1^∞ [s_n (W[K])]^P < ∞ or, expressed alternatively, W[K] belongs to the Schatten class C_p. The classical Wiener-Lévy Theorem is obtained as a corollary in the special situation when p = 1 and K(x,y) ≡ f(x-y) is a difference kernel. For the case 1 < p < 2 we generalize a Fourier series result of L. Alpar to the following theorem in integral operator theory. Theorem: Let W(z) be analytic (but not necessarily single-valued) in a region R and let K(x,y) satisfy the following conditions: a) K(x,y) maps the square [-π,π] x [-π,π] continuously into R. b) K(x,y) is 2π-periodic in x (or y). c) K(x,y) satisfies an integrated Lipschitz condition of order α relatively uniformly in x (or y) where p⁻¹ < α ≤ 1, 1 < p < 2. Then, if W(z) returns to its initial determination after z travels completely around any curve C_y (or C_x) of the form z = K(x,y), -π ≤ x (or y) ≤ π, then W[K(x,y)] ε C_p. To round out the paper, we show that analytic transformations preserve smoothness conditions of Lipschitz and bounded variation type, and consequently we are able to give a number of sufficiency conditions for analytic functions of kernels to be in various kernel classes. Finally, we investigate the converse of the Wiener-Lévy Theorem and how analogues of it relate to integral operators. The paper concludes with suggestions of several interesting questions warranting further study. / Ph. D.

The motivating idea behind the thesis is the study of the relationship of the degrees of the vertices of a graph and the structure of a graph. To each graph (indirected, without multiple edges) one can associate a graphical sequence by arranging the degrees of the vertices in their natural order. Conversely, an arbitrary sequence of numbers is graphical if it is a graphical sequence for some graph. In the first chapter general properties of graphical sequences are studied. We give conditions under which a sequence can be lengthened or shortened and have the property of being graphical be preserved. The concept of a 'transfer' is introduced to show how all realizations of a graphical sequence can be obtained from a given realization. Also in chapter one we show how graphical sequences can be used to characterize concepts like 'connected', 'block' and 'arbitrarily traceable'. If a graphical sequence has one 'realization' up to isomorphism then the sequence and the graph are called simple. Since simple graphs are determined up to isomorphism by the degrees of the vertices it is hoped that simple graphs will reveal in some measure the effect of the degree sequence on the structure of a graph. Thus, in chapter two, we attempt to characterize simple graphs--the central problem of the thesis. Simple trees, simple disconnected graphs, and simple graphs with cut points and no pendant vertices are characterized. (This means that characterizing simple blocks will solve the problem). Probably the most useful result is that a connected, simple graph must be of radius ≤ two and diameter ≤ three The third chapter is devoted to the problem of counting the number of non-isomorphic realizations of a given graphical sequence. Generating functions are used and several interesting special cases are given. These latter are in turn used to establish certain bounds on the number of realizations for sequences of a given length. / Ph. D.

Let S be a semigroup with zero and let a S\O. Denote by V(a) the set of all inverses of a, that is, V(a) = (x ∈ S: axa=a. xax=x). Let n be a fixed positive integer. A semigroup S with zero is said to be homogeneous n regular if the cardinal number of the set V(a) of all inverses of a is n for every nonzero element a in S. Let T be a subset of S. We denote by E(T) the set of all idempotents of S in T. The next theorem is a generalization of R. McFadden and Hans Schneider's theorem [1] . Theorem 1. Let S be a 0-simple semigroup and let n be a fixed positive integer. Then the following are equivalent. (i) S is a homogeneous n regular and completely 0-simple semigroup. (ii) For every a≠0 in S there exist precisely n distinct nonzero elements (xᵢ)_{i [= symbol with an n on top]l} such that axᵢa=a for i=1, 2, ..., n and for all c, d in S cdc=c≠0 implies dcd=d. (iii) For every a≠0 in S there exist precisely h distinct nonzero idempotents (eᵢ)_{i [= symbol with an h above]l} Eₐ and k distinct nonzero idempotents (fⱼ)_{j[= symbol with a k above]}= Fₐ such that eᵢa=a=afⱼ for i =1, 2, …, h, j = 1, 2, …, k hk=n, Eₐ contains every nonzero idempotent which is a left unit of a, Fₐ contains every nonzero idempotent which is a right unit of a and Eₐ ⋂ Fₐ contains at most one element. (iv) For every a≠0 in S there exist precisely k nonzero principal right ideals (Rᵢ)_{i[= symbol with a k above]1} and h nonzero principal left ideals (Lⱼ)_{j[= symbol with h above]1} such that Rᵢ and Lⱼ contain h and k inverses of a, respectively, every inverse of a is contained in a suitable set Rᵢ ⋂ Lⱼ for i = 1, 2, .., k, j = 1, 2, .., h and Rᵢ ⋂ Lⱼ for i = 1, 2, .., k, j = 1, 2, .., h, and Rᵢ ⋂ Lⱼ contains at most one nonzero idempotent, where hk = n. (v) Every nonzero principal right ideal R contains precisely h nonzero idempotents and every nonzero principal left ideal L contains precisely k nonzero idempotents such that hk=n, and R⋂L contains at most one nonzero idempotent. (vi) S is completely 0-simple. For every 0-minimal right ideal R there exist precisely h 0-minimal left ideals (Li)_{i[= symbol with an h above]1} and for every 0-minimal left ideal L there exist precisely k 0-minimal right ideals (Rj)_{j[= symbol with a k above]1} such that LRⱼ=LiR=S, for every i=1,2,..,h, j=l,2,.. ,k, where hk=n. (vii) S is completely 0-simple. Every 0-minimal right ideal R of S is the union of a right group with zero G°, a union of h disjoint groups except zero, and a zero subsemigroup Z uhich annihilates the right ideal R on the left and every 0-minimal left ideal L of S is the union of a left group with zero G’° a union of k disjoint groups except zero, and a zero subsemigroup Z' which annihilates the left ideal L on the right and hk=n. (viii) S contains at least n nonzero distinct idempotents, and for every nonzero idempotent e there exists a set E of n distinct nonzero idempotents of S such that eE is a right zero subsemigroup of S containing precisely h nonzero idempotents, Ee is a left zero subsemigroup of S containing precisely k nonzero idempotents of S, e (E(S)\E) = (0) = (E(S)\E)e, and eE⋂Ee = (e), where hk=n. S is said to be h-k type if every nonzero principal left ideal of S contains precisely k nonzero idempotents and every nonzero principal right ideal of S contains precisely h nonzero idempotents of S. W. D. Munn defined the Brandt congruence [2]. A congruence ρ on a sernigroup S with zero is called a Brandt congruence if S/ρ is a Brandt semigroup. Theorem 2. Let S be a 1-n type homogeneous n regular and complete:y 0-simple semigroup. Define a relation ρ on S in such a way that a ρb if and only if there exists a set (eᵢ) _{i[=symbol with an n above]1} of n distinct nonzero idempotents such that eᵢa=ebᵢ≠0, for every i=1, 2, . , n. Then ρ is an equivalence S\0. If we extend ρ on S by defining (0) to be ρ-class on S, then ρ is a proper Brandt congruence on S, then ρ ⊂ σ. Let P=(pᵢⱼ) be any n x n matrix over a group with G°, and consider any n distinct points A₁, A₂, . , A_n in the plane, which we shall call vertices. For every nonzero entry pᵢⱼ≠0 of the matrix P, we connect the vertex Aᵢ to the vertex Aⱼ by means of a path [a bar over both AᵢAⱼ] which we shall call an edge (a loop if i = j) directed from Aᵢ to Aⱼ. In this way, with every n x n matrix P can be associated a finite directed graph G(P). Let S=M°(G;In,In;P) be a Rees matrix semigroup. Then the graph G(P) is called the associated graph of the semigroup S, or simply it is the graph G(P) of S. Theorem 3. A Rees matrix semigroup S=M°(G;In,In;P) is homogenous m² regular if the directed graph G(P) of the semigroup S is regular of degree m [3, p. 11]. / Doctor of Philosophy

In this paper we study the concept of cotopology in the areas of cocon·tinuous functions and cocompact spaces. Initially we investigate and provide needed results concerning closed bases for a topological space. We then study cocontinuous functions by relating them to various other weaker forms of continuous functions, namely c-continuous, almost continuous and weakly continuous. We show that if (Y,U) is locally compact T₂, then f:(X,T)-->(Y,U) is cocontinuous if and only if f⁻¹(0) ε T for every 0 ε U such that (Y - 0) is compact. We note that every almost continuous function is cocontinuous, and we provide conditions under which a weakly continuous function is cocontinuous. We also show that a cocontinuous function from a saturated space to a regular space is continuous. In the area of cocompact spaces we first provide a partial answer to a question of J. M. Aarts as to when the union of cocompact subsets of a space is cocompact. We show that the union of a closed cocompact subset and a closed compact subset is cocompact. We then introduce the properties, locally cocompact and somewhere cocompact, and relate them to property L which was introduced by R. McCoy. We show that every somewhere cocompact regular space has property L, and that every locally cocompact regular space has property L locally. We provide examples to show that neither cocompact nor locally cocompact is equivalent to property L. / Ph. D.

A new approach to Kneser's Theorem, which achieves a simplification of the analysis through the introduction of maximal sets, the basic sequence of maximal e-transformations, and the limit set, B*, is presented. For two sets of non-negative integers, A and B, with C∈A⋂B, the maximal sets, Aᴹ and Bᴹ, are the largest supersets of A and B, respectively, such that Aᴹ + Bᴹ = A + B. By shifting from A and B to Aᴹ and Bᴹ to initiate the analysis, the maximal properties of Aᴹ and Bᴹ are exploited to simplify the analysis. A maximal e-transformation is a Kneser e-transformation in which the image sets are maximized in order to preserve the properties of maximal sets. The basic sequence of maximal e-transformation is a specific sequence of maximal a-transformations which is exclusively used throughout the analysis. B* is the set of all non-negative elements of sM which are not deleted by any transformation in the basic sequence of maximal e-transformations. Whether or not B* = {O} divides the analysis into two cases. One significant result is that B* = {O} implies δ (A + B) = δ (A, B) where δ(A + B) is asymptotic density of A + B and δ (A, B) is the two-fold asymptotic density of A and B. The second major result describes the structure of A + B when δ(A + B) < δ(A, B). With B* ≠ {0} it is shown, using only elementary properties of greatest common divisor and residue classes, that there exists C⊆ A+ B, 0εC, such that δ(C) ≥ δ(A, B) -1/g where g is the greatest common divisor of B* and C is asymptotically equal to C^(g), the union of all residue classes, mod g, which have a representative in C. The existence of C provides the crucial step in obtaining an equivalent form of Kneser’s Theorem: If A and B are two subsets of non-negative integer, 0εA⋂B, and δ(A + B) < δ(A, B), then there exists a positive integer g such that A + B is asymptotically equal to (A + B)^(g) and δ(A + B) = δ ((A + B)^(g)) ≥ δ (A^(g) , B^(g)) - 1/g ≥ δ(A, B) -1/g. / Ph. D.

If X is a complex with a triangulation and if P is a homeomorph of a polyhedron in X with respect to this triangulation, then P is tame in X if there is a homeomorphism h of X onto itself and another triangulation of X in which h(P) is a polyhedron. A function from one complex X into a complex is called tame and is said to preserve tame sets if for each tame set PcX, f(P) is tame. Tame local homeomorphisms from triangulated n-manifolds into triangulated n-manifolds and tame light open maps of 2-manifolds into themselves are homeomorphisms. Connected complexes are compact if and only if every tame map of the complex into itself has a polyhedral image. Tame linear maps of Euclidean spaces and tame simplicial maps on triangulated n-manifolds with boundaries are homeomorphisms if their images are of dimension greater than one. Functions from polyhedra into topological spaces which take tame arcs onto sets consisting of finite number of components have images of, at most, a finite number of components. If the function and its inverse takes tame sets onto tame sets then the image is connected, provided its image is in a complex. If the function is from a topological space into a polyhedron, then it is continuous if and only if its inverse takes tame arcs onto closed sets. Finally a function from a complex to a complex is continuous if its inverse takes tame sets onto tame sets. A function from an n-manifold into an n-manifold which has an image of dimension greater than one and which takes arcs onto arcs or points is a homeomorphism. A function from a compact triangulated n-manifold into a topological space which takes tame arcs onto arcs or points and whose image is not an arc or point is a homeomorphism. A function from a triangulated n-manifold into an n-manifold which takes tame arcs onto arcs or points and whose image is of dimension greater than one is a homeomorphism. A function from a triangulated n-manifold into a triangulated n-manifold which takes tame arcs onto connected tame sets such that the image of no tame arc contains a triod is a homeomorphism if its image set is not a point, arc or simple closed curve. Finally there are tame maps which raise the dimension of sets. And there are 1:1 maps which do not preserve tame sets. A K-R manifold is a n-manifold with boundary whose interior is Eⁿ and whose boundary is Eⁿ⁻¹. A 1:1 map of a 2-dimensional K-R manifold onto a 2-dimensional K-R manifold is a homeomorphism. / Doctor of Philosophy

Let X be a locally compact σ compact Hausdorff space. Let µ be a complete regular Borel measure defined on the Borel sets of X. It is shown that there is a base for the topology of X consisting of open sets whose boundaries are of µ measure zero. Let (S, p) be a metric space. It is shown that a function on X whose range is a subset of S can be uniformly approximated by µ almost everywhere continuous simple functions if, and only if, the function itself is µ almost everywhere continuous and its range is a totally bounded subset of S. S is then specialized to be a Banach algebra and several consequences are obtained culminating in the study of the ideal structure of the ring of ail µ almost everywhere continuous functions on X whose ranges are totally bounded subsets of a Banach algebra which is either the reals, complexes or quaternions. / Ph. D.

A topological space X is invertible at p ∈ X if for every· neighborhood U of p in X, there is a homeomorphism h on X onto X such that h(X - U) ⊆ U. X is continuously invertible at p ∈ X if for every neighborhood U of p in X there is an isotopy {h_t , 0 ≤ t ≤ 1, on X onto X such that h₁(X - U) ⊆ U. It is proved that, if X is a locally compact space which is invertible at a point p which has an open cone neighborhood, and if the inverting homeomorphisms may be taken to be the identity at p, then X is continuously invertible at p. A locally compact Hausdorff space X, invertible at two or more points which have open cone neighborhoods in X, is characterized as a suspension. A locally compact Hausdorff space X which is invertible at exactly one point p, which has an open cone neighborhood U such that U - p has two components, while X - p is connected, is characterized as a suspension with suspension points identified. Let Cⁿ be an n-conplex with invert point p. Let U be an open cone neighborhood of p in Cⁿ, and let L be the link of U in Cⁿ. Then it is shown that H_p(Cⁿ) is isomorphic to a subgroup of H_p-1(L). Invertibility properties of the i-skeleton of an n-complex are discussed, for i < n. Also, a method is described by which an n-complex which is invertible at certain points may be expressed as the union of subcomplexes, ca.ch of which is invertible at the same points. One-complexes with invert points are characterized as either a suspension over a finite set of points or a union of simple closed curves [n above ⋃ and i = 1 below that symbol], such that Sᵢ ⋂ Sⱼ = p, i ≠ j. It is proved that, if C² may be expressed as the monotone union of closed 2-cells. Also if the link of an open cone neighborhood of an invert point in a 2 - complex C² is planar, C² may be embedded in E³. / Doctor of Philosophy